Study uses Newton polytopes to distinguish Lagrangian fillings of Legendrian submanifolds.
problem Distinguishing Lagrangian fillings of Legendrian submanifolds.
method Utilizes Newton polytopes associated with augmented values of Reeb chords.
result Newton polytopes can distinguish infinitely many distinct Lagrangian fillings.
Study of measured laminations on surfaces using Newton polytopes and Poisson brackets.
problem Understanding the space of measured laminations on surfaces from a valuative perspective.
method Introducing Newton polytopes for character variety functions, defining tangent spaces, and identifying symplectic structures.
result Trace functions have unit coefficients at the extremal points of their Newton polytopes.
Proves necessity of at least log2(n) layers to compute maximum of n numbers.
problem Computing the maximum of n numbers with ReLU neural networks.
method Uses lattice polytopes and duality with Newton polytopes to prove depth lower bounds.
result Proves that log2(n) hidden layers are necessary and sufficient.
We study the Newton polytopes of determinants of square matrices defined over rings of twisted Laurent polynomials. We prove that such Newton polytopes are single polytopes (rather than formal differences of two polytopes); this result can be seen as analogous to the fact that determinants of matrices over commutative …
We show that link Floer homology detects the Thurston norm of a link complement. As an application, we show that the Thurston polytope of an alternating link is dual to the Newton polytope of its multi-variable Alexander polynomial. To illustrate these techniques, we also compute the Thurston polytopes of several speci…
In the classical theory of toric manifolds polytopes appear in two guises -- as Newton polytopes of line bundles on the complex, and as moment polytopes on the symplectic side, the link between the two being established by the prequantizability condition on the cohomology class of the symplectic form. Here we give a co…
We investigate Friedl-Lück's universal L2-torsion for descending HNN extensions of finitely generated free groups, and so in particular for Fn-by-Z groups. This invariant induces a semi-norm on the first cohomology of the group which is an analogue of the Thurston norm for 3-manifold groups. We prove…
Let M=G/H be a connected simply connected homogeneous manifold of a compact, not necessarily connected Lie group G. We will assume that the isotropy H-module g/h has a simple spectrum, i.e. irreducible submodules are mutually non-equivalent. There exists a convex Newton polytope N=N(G,H), which …
In this work, we examine the process of Tropical Polynomial Division, a geometric method which seeks to emulate the division of regular polynomials, when applied to those of the max-plus semiring. This is done via the approximation of the Newton Polytope of the dividend polynomial by that of the divisor. This process i…
Let f be an ordinary polynomial in C[z1,...,zn] with no negative exponents and with no factor of the form z1α1...znαn where αi are non zero natural integer. If we assume in addicting that f is maximally sparse polynomial (that its support is equal to the set of vertices of its Newton p…
We study invariant Einstein metrics on the indicated homogeneous manifolds M, the corresponding algebraic Einstein equations E, the associated with M and E Newton polytopes P(M), and the integer volumes ν=ν(P(M)) of it (the Newton numbers). We show that ν=80,152,...,152 respectively. It is claimed that…
The study broadens the concept of cyclic polytopes to Veronese polytopes.
problem Extending the framework of cyclic polytopes to a broader class of polytopes.
method Described facial structure and combinatorial characterisation of facets via σ-parity alternating sequences.
result Established a bijective correspondence between combinatorial types of Veronese polytopes and partitions of finite sets.
The paper studies deformation spaces of Coxeter truncation polytopes.
problem Understanding the geometric properties and deformations of Coxeter truncation polytopes.
method Analyzing Coxeter truncation polytopes and their deformation spaces.
result Description of deformation spaces for Coxeter truncation polytopes of dimension d⩾4. Neural networks approximate unit spheres as polytopes.
problem Approximating unit spheres with neural networks.
method Using ReLU activation in neural networks to generate polytopes.
result Neural networks can approximate unit spheres as polytopes.
The study classifies all compact hyperbolic polytopes with eight facets.
problem Classifying compact hyperbolic Coxeter four-polytopes with specific numbers of facets.
method Complete classification through mathematical analysis.
result The complete classification of compact hyperbolic Coxeter four-polytopes with eight facets.
Proves stability in Weyl polytopes using optimal transport.
problem Stability of Weyl polytopes under optimal transport.
method Optimal transport stability for reflexive Weyl polytopes.
result Weak metric SYZ conjecture holds for Delzant reflexive Weyl polytopes.
Faces of quasi-arithmetic Coxeter polytopes are also quasi-arithmetic.
problem Characterizing faces of quasi-arithmetic Coxeter polytopes.
method Proof of quasi-arithmetic property of faces and sufficient condition for arithmetic faces.
result Lower-dimensional faces of quasi-arithmetic Coxeter polytopes are quasi-arithmetic.
Given a finite collection P of convex n-polytopes in RP^n (n>1), we consider a real projective manifold M which is obtained by gluing together the polytopes in P along their facets in such a way that the union of any two adjacent polytopes sharing a common facet is convex. We prove that the real projective structure on…
This article announces the completion of the classification of rank 4 locally projective polytopes and their quotients. There are seventeen universal locally projective polytopes (nine nondegenerate). Amongst their 441 quotients are a further four (nonuniversal) regular polytopes, and 152 nonregular but section regular…
The study classifies all compact 5D polytopes with 9 facets.
problem Classifying compact hyperbolic Coxeter polytopes.
method Complete classification through mathematical analysis.
result A complete list of compact hyperbolic Coxeter 5D polytopes with 9 facets.
Smooth deformation space of Coxeter polytopes proven for orderable orbifolds.
problem Proving smoothness of deformation space for Coxeter polytopes.
method Analyzing natural map into realization space.
result Deformation space of Coxeter 3-polytopes is smooth.
Given a lattice L of R^n, a polytope D is called a Delaunay polytope in L if the set of its vertices is S\cap L where S is a sphere having no lattice points in its interior. D is called perfect if the only ellipsoid in R^n that contains S\cap L is exactly S. For a vector v of the Leech lattice Λ_{24} we define Λ_{24}(v…
The study classifies 331 specific 4D polytopes with 7 facets.
problem Classifying finite-volume hyperbolic Coxeter 4D polytopes.
method Complete classification through exhaustive search.
result 331 unique polytopes with 7 facets identified.
Contact manifolds' momentum polytopes are convex.
problem Understanding the structure of contact manifolds.
method Using isomorphism to toric varieties.
result Momentum polytopes of contact manifolds are convex.
Smooth approximations bound dihedral angles of convex polytopes.
problem Bounding dihedral angles of convex polytopes.
method Approximating polytopes with smooth hypersurfaces and using geometric relations.
result Established lower bounds on dihedral angles.
Polytopic Matrix Factorization models data as latent vectors from a polytope, maximizing determinant for identifiability.
problem Data decomposition with semi-structured latent vectors and polytope constraints.
method Model input data as latent vectors from a polytope, using determinant maximization for identifiability.
result Identifiability condition for polytopes with specific symmetry restrictions.
The article studies factorization structures in geometry and their applications to cones and polytopes.
problem Understanding and characterizing factorization structures in geometry.
method Comprehensive study of factorization structures, including structure theory, construction of compatible polytopes and cones, and derivation of generalised Gale's evenness condition.
result Established generalised Vandermonde identities and found examples of Delzant and rational Delzant compatible polytopes.
Examines nonrational polytopes and fans in toric geometry.
problem Understanding nonrational convex polytopes and fans in toric geometry.
method Discussion and interrelation of recent developments.
result Exploration of nonrational polytopes and fans in toric geometry.
New methods classify hyperbolic polytopes with up to 40 facets.
problem Classifying compact hyperbolic Coxeter polytopes with specific facet counts.
method New combinatorial method via point set order types.
result Proves existence of a compact hyperbolic Coxeter 29-polytope with at least 40 facets.
The paper explains how to parameterize facets of moment polytopes in real symplectic geometry.
problem Understanding the moment polytopes in real symplectic geometry.
method Parameterizing equations of facets of Delta(Z) in terms of real Ressayre's pairs of Z.
result Parameterization of facets of moment polytopes explained.
A new method for optimization in probability space using Newton's flows.
problem Optimization in probability space with information metrics.
method Information Newton's flows, including Fisher-Rao and Wasserstein-2 metrics, with Newton's Langevin dynamics and variational methods.
result Effective numerical implementation and convergence results for the proposed method.
We introduce two operations named biflip and puzzle-move on simple polytopes producing polytopes with diffeomorphic moment-angle manifolds.
Unified cosmological and Einstein polytope theories.
problem Unified understanding of cosmological and Einstein polytope theories.
method Unified combinatorial perspective of cosmological and Einstein polytope theories.
result Unified construction of cosmological and Einstein polytope theories.
The study of symmetries in manifolds derived from colored polytopes.
problem Existence and types of symmetries in rational homology 3-spheres.
method Analysis of hyperbolic manifolds and right-angled polytopes.
result Described how to create colorings with specific symmetries.
Accelerates Birkhoff projection for manifold-constrained hyper-connections with high accuracy and speed.
problem Inaccurate and slow Birkhoff projection in mHC implementations.
method Dual formulation, Newton's method, implicit differentiation, warp-level CUDA kernel.
result Substantial speedups and accuracy improvements in doubly stochastic projections.
Diffeomorphisms of convex polytopes form a Lie group.
problem Understanding transformations of convex polytopes.
method Forming a Lie group from diffeomorphisms of convex polytopes.
result The group of diffeomorphisms of a convex polytope is a regular Lie group.
Efficiently projects points onto polytopes, especially useful in web-scale applications.
problem Efficiently projecting points onto polytopes in large-scale applications.
method Developed a vertex-oriented incremental algorithm for polytope projection, tailored for simplex and unit-box cut polytopes.
result Majority of projections lie on vertices of polytopes, leading to significant performance improvements.
Geometric constraints help classify hyperbolic polytopes.
problem Classifying reflective anisotropic Lorentzian lattices and cocompact arithmetic hyperbolic reflection groups.
method Established geometric constraints on compact Coxeter polytopes in hyperbolic spaces.
result Geometric constraints are useful for classifying hyperbolic polytopes.
New noncompact Coxeter polytopes found in various dimensions.
problem Classifying and constructing noncompact hyperbolic Coxeter polytopes.
method Maximal-cusp density and noncompact analog of Bogachev-Douba-Raimbault's argument.
result Infinitely many pairwise incommensurable noncompact Coxeter polytopes in dimensions 4-9.
We describe stochastic Newton and stochastic quasi-Newton approaches to efficiently solve large linear least-squares problems where the very large data sets present a significant computational burden (e.g., the size may exceed computer memory or data are collected in real-time). In our proposed framework, stochasticity…
Proves weight polytope matches with energy vectors in toric varieties.
problem Understanding the relationship between weight polytopes and energy functionals in toric varieties.
method Combines two slope formulas of K-energy in the toric setting.
result Weight polytope of Hurwitz form matches with convex hull of characteristic vectors.
Unique floating and buoyancy surfaces identify convex polytopes.
problem Identifying convex polytopes from their flotation and buoyancy surfaces.
method Proving uniqueness of surfaces for polytopes with uniform or prescribed density.
result Floating and buoyancy surfaces uniquely determine convex polytopes.
Newton's method solves variational problems on manifolds.
problem Solving variational equations on manifolds.
method Newton's method with affine covariant damping strategy.
result Numerical results for variational problems demonstrated.
The study proves curvature rigidity for convex polytopes.
problem Proving curvature rigidity for convex polytopes.
method Using Fredholm theory for Dirac operators and a theorem of Fefferman and Phong.
result Scalar curvature rigidity theorem for convex polytopes proved.
The paper classifies compact hyperbolic Coxeter polytopes and improves upper bounds.
problem Classifying compact hyperbolic Coxeter polytopes and understanding their combinatorial properties.
method Study of imes0-products of Lannér diagrams, proving superhyperbolic properties, and analyzing Lannér subdiagrams. result Improved upper bounds on the dimension of compact hyperbolic Coxeter polytopes.
We study unbounded 2-dimensional metric polytopes such as those arising as Kähler quotients of complete Kähler 4-manifolds with two commuting symmetries and zero scalar curvature. Under a mild closedness condition, we obtain a complete classification of metrics on such polytopes, and as a result classify all possible m…
We investigate polyhedral 2k-manifolds as subcomplexes of the boundary complex of a regular polytope. We call such a subcomplex {\it k-Hamiltonian} if it contains the full k-skeleton of the polytope. Since the case of the cube is well known and since the case of a simplex was also previously studied (these are so…
We study the space of generalized translation invariant valuations on a finite-dimensional vector space and construct a partial convolution which extends the convolution of smooth translation invariant valuations. Our main theorem is that McMullen's polytope algebra is a subalgebra of the (partial) convolution algebra …